Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Atypical stars on a directed landscape geodesic
Speaker: Manan Bhatia, MIT
Date: March 20, 2023
Time: 18:30:00 Hours
Venue: via Zoom
Abstract: In random geometry, a recurring theme is that all geodesics emanating from a typical point merge into each other close to their starting point, and we call such points as 1-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this talk, we will consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virg and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. The main result we will discuss is that the above dimension is almost surely equal to 1/3. This is in contrast to Ganguly-Zhang where it was shown that the set of atypical stars on the line {x=0} has dimension 2/3. This reduction of the dimension from 2/3 to 1/3 yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.
Title of Talk: Atypical stars on a directed landscape geodesic
Speaker: Manan Bhatia, MIT
Date: March 20, 2023
Time: 18:30:00 Hours
Venue: via Zoom
Abstract: In random geometry, a recurring theme is that all geodesics emanating from a typical point merge into each other close to their starting point, and we call such points as 1-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this talk, we will consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virg and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. The main result we will discuss is that the above dimension is almost surely equal to 1/3. This is in contrast to Ganguly-Zhang where it was shown that the set of atypical stars on the line {x=0} has dimension 2/3. This reduction of the dimension from 2/3 to 1/3 yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.